Tail probabilities of low-priority waiting times and...

Queueing Systems 34 (2000) 215–236 215

Tail probabilities of low-priority waiting times andqueue lengths in MAP/GI/1 queues ∗

Vijay Subramanian a,∗∗ and R. Srikant b

a Mathematics of Communication Networks, Motorola, 1501 W. Shure Drive, Arlington Heights,IL 60004, USA

E-mail: [email protected] Coordinated Science Laboratory and Department of General Engineering, University of Illinois,

1308 W. Main Street, Urbana, IL 61801, USAE-mail: [email protected]

Received 15 January 1999; revised 19 August 1999

We consider the problem of estimating tail probabilities of waiting times in statisticalmultiplexing systems with two classes of sources – one with high priority and the other withlow priority. The priority discipline is assumed to be nonpreemptive. Exact expressionsfor the transforms of these quantities are derived assuming that packet or cell streams aregenerated by Markovian Arrival Processes (MAPs). Then a numerical investigation of thelarge-buffer asymptotic behavior of the the waiting-time distribution for low-priority sourcesshows that these asymptotics are often non-exponential.

Keywords: priority queues, tail probabilities, waiting times, Markovian arrival processes

1. Introduction

In this paper, we consider a MAP/GI/1 queue with two priority classes operatingunder a non-preemptive priority service discipline, and we are interested in the behaviorof the waiting time of the low priority arrival. MAPs are very commonly used todescribe arrival processes in high-speed networks. We assume that the service-timedistribution is the same for both priority classes, a special application of this wouldbe in ATM networks where the cell size is a constant 53 bytes. Since the models weconsider fall under the class of M/G/1 type models defined by Neuts, we would drawupon the substantial literature on such models in [24,25,27,31].

Our main results are follows. We derive exact expressions for the transforms ofthe waiting time and queue length of the low-priority sources. The transforms couldbe used to obtain the exact tail probability of these quantities using Laplace transforminversion techniques. We also numerically investigate the asymptotic behavior of thetail probabilities. For our numerical studies, we focus on the tail behavior of the low-

∗ Research supported in part by NSF Grant NCR-9701525.∗∗ The first author was previously with the University of Illinois.

J.C. Baltzer AG, Science Publishers

216 V. Subramanian, R. Srikant / Tail probabilities in priority queues

priority waiting time. Our numerical results suggest that the tail of the low-prioritywaiting time distribution is often non-exponential. An inspiration for our work is theresults of Abate and Whitt [4], who showed the non-exponential asymptotic behaviorfor the low-priority sources for the case when the arrival processes for both classesare Poisson. Our results generalize this to MAP/G/1 queues and provide exact ex-pressions for the transforms of the low priority waiting times. While the results in [4]show the exact nature of the asymptote, we present strong numerical evidence to onlydemonstrate that the asymptote is non-exponential. In addition, also using numeri-cal results, we show that the behavior of the asymptote is significantly different inthe case of general MAP arrivals as compared to the case of Poisson arrivals studiedin [4]. Specifically, in the case with Poisson arrivals, the low-priority waiting-timedistribution has an exponential asymptote when the low-priority arrival rate is abovea certain threshold. This is not necessarily true in the case with MAP arrivals. Otherdifferences are pointed out in section 4.

An important application of tail probability computations is in the design ofadmission control schemes for high-speed networks, where admitted sources are guar-anteed a prespecified Quality-of-Service (QoS). QoS is typically specified in terms ofquantities such as probability of cell loss, upper limits on waiting times, etc. A pop-ular technique is to convert these QoS requirements into a single number called theeffective bandwidth and use this quantity very much like a bandwidth requirementin circuit-switched networks. The effective bandwidth approximation relies on largedeviations results of the following form:

limB→∞

1B

logP (X > B) = −δ, (1)

where X is the workload and δ > 0; see, for example, [16,23,36]. Thus, given a finitebuffer size B, a natural approximation to the loss probability P (X > B) is

P (X > B) ≈ e−δB . (2)

Similar large deviations results and approximations are also available for waiting timesand queue lengths [36]. In ATM networks, the cell size is constant. A common ap-proximation to this is to use a Erlang service-time distribution with many phases, oftenmaking the problem amenable to a matrix–geometric type analysis. This approxima-tion of deterministic service times by a phase-type distribution is discussed extensivelyin [11], but we do not address it here.

While there are several sources of error in using the approximation (2), it has beenshown that the approximation can be refined as P (X > B) ≈ α e−δB , where α canbe computed either numerically [11], or by capturing the gains of bufferless statisticalmultiplexing using the Chernoff bound [17] or appropriate large deviations scaling toaccount for a large number of sources [8,14]. Such a refinement is considered to bereasonably accurate for most practical applications. Further, for most well-behavedservice-time distributions, if α is exactly calculated, P (X > B) ∼ α e−δB [2], wheref (x) ∼ g(x) denotes limx→∞ f (x)/g(x) = 1. Assuming a preemptive resume model,

V. Subramanian, R. Srikant / Tail probabilities in priority queues 217

large deviations results of the form (2) can also be obtained for statistical multiplexerswith priorities [6,10], based on earlier work characterizing the large-deviations ratefunction of the output process of a single-class, single-server queue [9,15]. A moregeneral case has been considered in [7,29], where large-deviation asymptotics havebeen obtained for queue lengths and virtual waiting-times in two queues operatingunder the generalized processor-sharing (GPS) discipline. One of the contributions ofour paper is to point out that for the nonpreemptive priority model (which shouldbe very close to the preemptive resume model when the packet sizes are small), theasymptotic form P (X > B) ∼ α e−δB does not hold, in general, for the low-prioritysources.

Previously, Takine et al. [35] considered nonpreemptive priority queues withMarkovian arrivals and independent and identically distributed service times for eachpacket. They derived expressions for the mean waiting time of each packet class andalso presented algorithms for evaluating the mean waiting times. Takine et al. [34]derived expressions for the workload in a MAP/GI/1 queue with state-dependentarrivals. This was used to derive the Laplace–Stieltjes transform (LST) for the waitingtime of packets in each class of a MAP/GI/1 queue with preemptive resume priority.Sugahara et al. [32] analyzed a two-priority (nonpreemptive priority) queue with thehigh class arrivals from a 2-state Markov Modulated Poisson process and the low classarrivals from a Poisson process. Using a supplementary variable technique for solvingthe partial differential equations for the system, they derived the joint probabilitygenerating functions of the stationary queue-length distributions and the LST of thestationary waiting time-distributions of high- and low-priority packets. Our derivationof these quantities use a different approach primarily relying on the properties of MAParrival processes and the key renewal theorem for Markov renewal processes [20] asin the derivation of the results in [26]. Further, we also extend the results to the casewhere both the high- and low-priority classes are MAPs using Palm theory [5].

Elwalid and Mitra [18] consider a multi-service multiplexing system with Markovmodulated fluid inputs and derive approximations to the tail probabilities of the queue-length distributions. The main idea there is an elegant Markovian approximation for thebusy-period of the high-priority packets. The Markovian approximation then naturallyprovides a exponential tail for the low-priority buffer occupancy, which is used in anadmission control scheme for such a multiplexer. Since our model is not a fluid model,the results in [18] do not directly apply. It may be possible to connect our results tothe results in [18] using appropriate fluid limits as in [12]. However, we do not pursuethis avenue of research in this paper. Using the results in [18], Presti et al. [30] haveobtained approximations for the more general case of GPS service discipline.

The rest of this paper is organized as follows. In section 2, we introduce MAPsand present some known properties of MAPs and MAP/GI/1 queues that will beused later. Priority queues are considered in section 3 and an exact expression forthe transform of the waiting-time distribution of the low-priority packets is derived.Section 4 presents numerical results and discusses the conclusions from these re-sults.


2. MAPs and MAP/GI/1 queues

A MAP is a continuous-time Markov chain described by the following generator:

Q =

D0 D1 0 0 . . .

D0 D1 0 . . .

D0 D1. . .

. . . . . .

,

where D ≡ D0 + D1 with D 6= D0, is also a generator. We assume that D isirreducible. One can associate an arrival process with this Markov chain as follows:an arrival occurs whenever there is a state transition into a state corresponding to a D1

block and there is no arrival otherwise. D0 and D1 are m×m matrices, where m is thenumber of phases of the arrival process. This is a special case of versatile Markovianpoint processes introduced by Neuts [27] and studied under various names [24,25,31],most commonly referred to as the Batch Markovian Arrival Processes (BMAPs). Wehave simply restricted our attention to the case where the batch size at each arrivalpoint can be at most 1. We will not discuss all the properties of MAPs here, the readeris referred to [25] for an introduction. However, before we introduce priority queues,we present some well-known properties of MAPs and MAP/GI/1 queues which willbe useful for our analysis later. These results can be found in [24–26,31].

If we let N (t) be the number of arrivals in (0, t] and J(t) be the phase of thesystem at time t, then (N (t),J(t)) is a Markov chain on the state space {(n, j), n > 0,j = 1, 2, . . . ,m} with generator Q. Define D(z) to be

D(z) ≡ D0 +D1z for |z| 6 1,

and let Pa(n, t) be an m×m matrix denoting the number of arrivals in (0, t] with the(i, j)th element defined as[

Pa(n, t)]ij≡ Prob

(N (t) = n, J(t) = j | N (0) = 0, J(0) = i

).

Then, the matrix generating function P ∗(z, t) of the number of arrivals in [0, t]1 isgiven by

P ∗(z, t) = eD(z)t, for |z| 6 1, t > 0,

where P ∗(z, t) =∑∞

n=0 Pa(n, t)zn. Let π be the stationary distribution of D. Thenthe steady-state mean arrival rate is given by

λ = πD1e,

and since D is irreducible, this is also the long-term arrival rate.

1 Most quantities in matrix-analytic theory are matrices such as Pa(n, t) and P ∗(z, t) with each elementcorresponding to a particular phase transition. However, for convenience, we will often refer toquantities such as “the number of arrivals in [0, t]” without referring to the phase transition, but thisshould not cause confusion.


In the rest of this section, we state a few lemmas listing some properties ofMAP/GI/1 queues that will be used later. Let H be the service-time distribution withLST h (i.e, h(s) =

∫∞0 e−st dH(t)) and mean 1/µ with µ > 0. The traffic intensity at

the MAP/GI/1 queue is given by ρ = λ/µ.

Lemma 1. Let A(z, s) be the 2-dimensional transform of the number of arrivals andtime interval between successive departure epochs given that the first departure didnot leave the system empty. A(z, s) is given by

A(z, s) = h(sI −D(z)

), where Re(s) > 0 and |z| 6 1.

In the previous lemma h(sI − D(z)) is a scalar function evaluated at a matrixargument. This is evaluated in the standard manner by substituting the matrix argumentin the power series expansion of h. This same interpretation will be used in the restof the paper whenever a scalar function is evaluated at a matrix argument. For moredetails, the reader is referred to [25, section 5.1].

Lemma 2. We have the following equation for the 2-dimensional transform of thenumber served during, and the duration of, the busy period

G(z, s) = zh(sI −D

[G(z, s)

]).

Lemma 3. Define G ≡ G(1, 0). The matrix G has the following properties:

1. G is stochastic whenever ρ < 1. Let g be the invariant probability vector of G.

2. The jth component of vector g is the stationary probability that the arrival processis in phase j at times of departures given that the system is empty.

Lemma 4. Define A(z) ≡ A(z, 0) = h(−D(z)). The queue-length generating functionof a MAP/GI/1 queue at an arbitrary time is given by

Y (z) = (1− ρ)g(z − 1)A(z)[zI −A(z)

]−1. (3)

3. Priority MAP/GI/1 queues

In this section, we derive the transforms of various quantities of interest forMAP/GI/1 queues with two priority classes where the classes have separate buffers.Throughout, we assume that the service-time distribution is the same for both classes.For ease of exposition, we first consider the case where the low-priority arrivals arePoisson. Later, we generalize this to the case where both classes generate arrivalsaccording to MAPs.


3.1. Poisson low-priority arrivals

Let the low-priority arrivals be Poisson with rate λl. The high-priority arrivalprocess is a MAP with Dh

0 and Dh1 as the relevant parameters and the rate is de-

noted as λh. The traffic intensity is given by ρl ≡ λl/µ and ρh ≡ λh/µ withρ ≡ ρh + ρl < 1 for stability. Note that the low-priority class can be considered asa MAP with parameters Dl

0 = −λl and Dl1 = λl. Define Dh(z) ≡ Dh

0 + Dh1 z and

DS(z) ≡ Dh0 + Dh

1 z + zλlI − λlI . Then, lemmas 1–4 can be applied to the singleclass MAP/GI/1 obtained by considering the two buffers (high and low priority) to-gether, and the arrival process being the superposition of the high- and low-priorityarrival processes. As in DS(z), throughout we will use the subscript S to denote thissingle-priority-class MAP/GI/1.

We first derive an expression for the LST of the waiting-time distribution of thelower priority packets. Since the low-priority and high-priority packets have the sameservice-time distributions, we only need to monitor the total number of packets in thesystem. We first note two useful facts:

• The low-priority arrival process is Poisson and, thus, we use the PASTA prop-erty [37] to claim that upon arrival a low-priority packet sees YS(z)e as the distri-bution of the number of packets in service, where YS(z) is given by (3) with D(z)replaced by DS(z), and e is a vector with all elements equal to 1.

• The remaining service of the packet in service (if any) is given by the station-ary excess distribution associated with H (again by PASTA). Denote the resid-ual service-time distribution by R and its LST by r, where R is given byR(t) = µ

∫ t0 (1−H(u)) du and, thus, r(s) = (1− h(s))µ/s.

Now we proceed along the lines of the derivation of the Takacs equation andthe Kendall functional equation for the busy period of the M/G/1 queue [13; 19,chapter XIV]. Specifically, we note that if a lower-priority packet sees n packets (ofboth classes together) ahead of it, then its waiting time is independent of the orderin which all the packets ahead of it (and any high-priority arrivals before it beginsservice) are served. Thus, the waiting time is composed of

• W1: The service times of the packet in service and all high-priority packets thatarrive during the service of the packet currently in service.2

• W2: The n− 1 high-priority busy periods generated by the n− 1 packets ahead ofit in the queue.3

The LST of W1 can be easily seen to be

Gl(s) = r(sI −Dh

[Gh(s)

]).

2 Note that we need to consider the phase of the arrival process at the beginning of the service and atthe end. Thus, W1 is actually a matrix-valued random variable.

3 Again we need to take into account of the arrival process phases. Thus, W2 is also a matrix-valuedrandom variable.


From lemma 2, the high-priority busy period is given by

Gh(s) = h(sI −Dh

[Gh(s)

]). (4)

Therefore, using the relationship between r(s) and h(s), Gl(s) can be rewritten as

Gl(s) = µ(I −Gh(s)

)[sI −Dh

[Gh(s)

]]−1. (5)

Now we are ready to state and prove the following theorem.

Theorem 5. The LST of the complementary cumulative distribution function (ccdf)of the low-priority waiting time is given by

wCl (s) =1− wl(s)

s,

where

wl(s) = (1− ρ) +{YS(Gh(s)

)− (1− ρ)gS

}G−1h (s)Gl(s)e. (6)

Proof. Let yS(n) be distribution of the total number of customers in the system, i.e.,the inverse transform of YS(z) given by (3) with D(z) replaced by DS(z). In otherwords, YS(z) =

∑∞n=0 yS(n)zn. Thus, yS(n) is a vector such that each element of the

vector corresponds to an arrival phase and gives the probability that the total number(sum of high-priority and low-priority packets) is n. Therefore, from the discussionprior to the theorem, we have that the LST of the waiting-time distribution of thelow-priority packets wl(s) is given by

wl(s) = (1− ρ) +∑n>0

yS(n)Gl(s)Gn−1h (s)e, (7)

where we have used the fact that the waiting time is zero when a packet arrives whenthe system is empty, which occurs with probability 1−ρ. From equation (4) we deducethat Gh(s) is a power series in Dh[Gh(s)]. From equation (5) we can deduce that evenGl(s) is a power series in Dh[Gh(s)]. Thus, Gl(s) and Gh(s) commute. Using thiswe have the following form for wl(s):

wl(s) = (1− ρ) +

{∑n>0

yS(n)Gnh(s)

}G−1h (s)Gl(s)e

= (1− ρ) +{YS(Gh(s)

)− (1− ρ)gS

}G−1h (s)Gl(s)e. (8)

For the last equation above, we have used the interpretation of gS from lemma 3. �

The next result presents the transform of the stationary distribution of number ofhigh-priority and low-priority packets in the system, the proof of which is providedin the appendix. By PASTA, this is also the same as the distribution of number ofhigh-priority and low-priority packets seen by a low-priority packet arrival.


Theorem 6. Let y(n,m) be a vector such that its ith component is the proba-bility that there are n high-priority and m low-priority packets in the system,and the arrival process is in phase i. Its 2-dimensional transform Y (z,w) =∑∞

n=0

∑∞m=0 y(n,m)znwm is given by

TY (z,w) =P (0, 0)(−(Dh

0 − λlI)−1)(

zDh1 + wλlI − I

)(I −A(z,w)

)(−D(z,w)−1)

+ P (0, 0)[−(I −A(z, 0)

)(−D(z, 0)−1)+

(−(Dh

0 − λl)−1)]

+ P (z,w)(I −A(z,w)

)(−D(z,w)−1)

+ P (z, 0)(I −A(z, 0)

)(−D(z, 0)−1), (9)

where

T =1

λh + λl, D(z,w) = Dh

0 − λlI +Dh1 z + λlIw,

P (z,w) =

{P (0,w)

(z

w− 1

)+ P (0, 0)

z

w

[−w(Dh

0 − λlI)−1(

Dh1 + λlI

)− I]}

×A(z,w)(zI −A(z,w)

)−1, (10)

and

A(z,w) =

∫ ∞0

e(Dh0 −λlI+Dh1 z+λlIw)x dH(x), (11)

P (0,w) =Q(w)(P (0, 1)e

), (12)

Q(0) =P (0, 0)

(P (0, 1)e), (13)

Q(w) =Q(0){−w(Dh

0 − λlI)−1(

Dh1 + λl

)− I}GH (w)

(wI −GH (w)

)−1, (14)

GH (w) =Gh(λl − λlw), (15)

P (0, 0) =1− ρ

λh + λlIgS(−Dh

0 + λlI), (16)

P (0, 1)e=λl

λh + λl+

1− ρλh + λl

gSDh1 e. (17)

We can now write out the expression for Y C(w) using the expression for Y (z,w) andthe following relation:

Y C(w) =1− Y (1,w)e

1− w . (18)

3.2. MAP arrival process for the low-priority class

Here we let the high-priority class have Dh0 , Dh

1 as the matrices associated withthe arrival process and, similarly, let Dl

0, Dl1 be the matrix parameters of the low-


priority arrival process. We still assume that the service-times are distributed identicallyfor the two classes. Let Ih, Il denote identity matrices of the same order as Dh

1 , Dl1,

respectively.From approaches standard in matrix-analytic theory, we have the following the-

orem which is the equivalent of theorem 6. The differences between the proofs of thetwo theorem is outlined in the appendix. Note that we only need to make the appropri-ate changes to the renewal function of the Markov process, the distribution of arrivals,and the distribution of customers at departure epochs in the proof of theorem 6 givenin the appendix. While theorem 6 also gives the transform of the distribution seenby low priority arrivals when they are Poisson, the following theorem gives only thestationary distribution.

Theorem 7. Let y(n,m) be a vector such that its ith component is the probabilitythat there are n high-priority and m low-priority packets in the system, and the arrivalprocess is in phase i. Its 2-dimensional transform

Y (z,w) =∞∑n=0

∞∑m=0

y(n,m)znwm

is given by (9) with the following modifications:

(i) Dh0 replaced by Dh

0 ⊗ Il,(ii) −λlI replaced by Ih ⊗Dl

0,

(iii) Dh1 replaced by Dh

1 ⊗ Il,(iv) λlI replaced by Ih ⊗Dl

1,

(v) Dh(z) = Dh0 ⊗ Il +Dh

1 ⊗ Ilz,

(vi) D(z,w) = Dh0 ⊕Dl

0 +Dh1 ⊗ Ilz + Ih ⊗Dl

1w,

(vii) DS(z) = D(z, z), and

(viii) GH(w) solves

GH (w) =A(GH(w),w

)=

∫ ∞0

e(Dh0 ⊗Il+Ih⊗Dl0+(Dh1 ⊗Il)GH (w)+Ih⊗Dl1w)x dH(x)

= h(−Dh

0 ⊗ Il − Ih ⊗Dl0 −

(Dh

1 ⊗ Il)GH (w)− Ih ⊗Dl

1w). (19)

In the rest of this subsection we provide expressions for the distribution of thenumber of packets seen by a low-priority arrival and the waiting-time distributionof the low-priority packets, i.e., wl(s). Since the stationary distributions remain thesame as before, we only need to derive the relevant customer (packet) averages. Letπl be the stationary distribution of the state of the low-priority arrival process. Theproofs of these are essentially along the lines of [26, theorem 9] where the relationship


between time averages and customer averages in [33] is generalized to MAP/GI/1queues.

Theorem 8. Let Y aS (z) be the z-transform of the distribution of the total number of

packets in the system seen by a low-priority arrival. Y aS (z) is given by

Y aS (z) = YS(z)

(Ih ⊗Dl0)e

πlDl0el

, (20)

where e is a column vector of ones of size given by the size of Ih ⊗Dl0 and el is a

column vector of ones with size the same as Dl0. Thus, the ccdf has a z-transform

given by

Y aS (z) =

1− Y aS (z)

1− z . (21)

Theorem 9. The 2-dimensional transform of the number of high-priority and low-priority packets seen by a low-priority arrival, denoted by Ya(z,w) is given by

Y a(z,w) = Y (z,w)(Ih ⊗Dl

0)e

πlDl0el

, (22)

where Y (z,w) is given in theorem 7. Therefore, the tail of the distribution of thenumber of low-priority packets seen by a low-priority arrival has a z-transform givenby

Ya(w) =1− Ya(1,w)

1−w . (23)

Theorem 10. The virtual waiting-time distribution of the low-priority arrivals is givenby

W lV (s) = (1− ρ)gS +

∑n>0

yS(n)Gl(s)Gn−1h

= (1− ρ)gS +{YS(Gh(s)

)− (1− ρ)gS

}G−1h (s)Gl(s). (24)

Theorem 11. The waiting-time distribution of the low-priority arrivals is given by

wl(s) = W lV (s)

(Ih ⊗Dl0)e

πlDl0el

. (25)

Thus, the LST of the low-priority waiting time ccdf is given by

wCl (s) =1− wl(s)

s. (26)


4. Asymptotics of the tail probabilities

In this section, we present results to show that the low-priority tail asymptoticsfor waiting times are non-exponential. Assuming an asymptote of the form

P (W > T ) ∼ αT−β e−δT , (27)

we present numerical evidence to show that often β 6= 0. For general MAPs, it isdifficult to obtain closed-form expressions for α, β and δ. Therefore, we computethese numerically using the transform expressions obtained in the earlier section. Forthis purpose, we use the techniques in [1]. However, to use these numerical results,one has to first hypothesize a form for the asymptotics and, then, proceed to obtain theparameters of the asymptotic form. Therefore, we first present a simple model whichsuggests asymptotics of the form (27). This example proving the asymptotic form isnot for MAP processes but it is for fluid models which are closely related to MAPswhen the service times are small, as shown in [12]. However, it should be notedthat nonpreemptive and preemptive priorities are the same in fluid models, and thus,we lose the detailed structure of the MAP queueing system when considering the fluidmodel. Thus, this example only suggests the asymptotics, but is not a conlcusive proofthat the asymptote is indeed of the assumed form. Nevertheless, it serves as a simpleexample to illustrate why one might suspect non-exponential asymptotic behavior forthe low-priority tail probability.

Our principal conclusion from the numerical study does not depend on the as-sumption that the asymptotic form is given by (27). Our principal conclusion is simplythat the asymptote is non-exponential, which is arrived at by showing that β convergesto a nonzero value. Hence, the assumption of the asymptotic form is not importantto reach this conclusion. But the fluid-model example and the results in [4] suggestthe form (27), and, thus, it is interesting to study whether the parameter β behaves asit does for the case of Poisson arrivals studied in [4]. The numerical evidence willshow that, in general, the behavior is different with general MAP arrivals, i.e., eitherthe asymptotic form is not of the type given in (27), or, if it is, then the behavior ofβ is different.

4.1. Fluid model example

As mentioned earlier, non-exponential tails for waiting times of low-priority wait-ing times was shown in [4] for M/GI/1 models. The key idea was to use a geometricrandom sum representation for the transform of the low-priority waiting times wherethe stationary excess of the “service time” of the low-priority packets includes theeffect of the busy period due to the high-priority sources. The example we providehere complements the results in [4] by considering a high-priority source that is anon–off Markov-modulated rate process which is commonly used in modeling arrivalprocesses. However, as we shall see, our example which considers the tail of thelow-priority workload has a simpler direct interpretation in terms of the busy period of


the M/GI/1 queue. Since we are considering a fluid model, it is simpler to considerworkload, instead of waiting times, and hence, we do so.

Consider a multiplexer with two buffers. Buffer 1 is fed by source 1 that alternatesbetween on and off states. It spends an exponentially distributed amount of time ineach state with the mean on and off times given by 1/qd and 1/qu, respectively.When source 1 is on, it produces fluid at rate λ1. Source 2 is a constant rate sourceproducing fluid at rate λ2. The server rate is c and its service discipline is generalizedprocessor sharing (GPS) with weights φ1 and φ2 for sources 1 and 2, respectively [28].Comparing this to our MAP models earlier, this would be the fluid-equivalent of a two-state Markov-modulated Poisson process (MMPP) for source 1 and a Poisson processfor source 2 with the service times being a Erlang-k distribution for large k. Note thatthe choice of φ2 = 0 leads to a strict priority scheme with lower priority for source 2as we had been considering in the previous sections. But here we let φ2 be possiblynonzero to allow us to study the more general case of GPS.

We assume that

λ1 >φ1c

φ1 + φ2and λ2 >

φ2c

φ1 + φ2.

We will denote service rate guaranteed to source i as gi where gi = cφi/(φ1 + φ2).Clearly, when λ2 < g2, buffer 2 is always empty. In what follows, we show thatone can exactly determine the asymptotic form of the workload in buffer 2 and whenλ2 > g2, this asymptotic form is non-exponential. For stability, we assume that(

ququ + qd

)λ1 + λ2 < c.

We first note the following two facts that we will use:

• Due to our assumptions on the arrival rates λ1 and λ2, whenever source 1 is eitheron or back-logged, it receives service at rate g1, and source 2 receives service atrate g2.

• When buffer 1 is empty, source 2 receives service at rate c when it is back-loggedand at rate λ2 when its buffer is empty.

A busy period of source 1 begins in the on state and ends when its buffer isempty. We denote the LST of its busy period as B(s). We derive an expression forB(s) by exploiting a connection between the fluid model and an appropriately definedM/GI/1 queue as in [21,22].

Let X1(t) be X2(t) be the workload in buffers 1 and 2, respectively. DefineY1(t) ≡ X1(t)/g1. Let tn be the begin time of the nth off period of source 1. Then,

Y1(tn) =[Y1(tn−1)− an

]++λ1 − g1

g1bn, (28)


Figure 1. (a) Actual busy period due to source 1. (b) Busy period considering only the off-times ofsource 1.

where an is the duration of the (n − 1)th off period and bn is the duration of the onperiod following the (n− 1)th off period. Further, for 0 6 δ < an+1,

Y1(tn + δ) =[Y1(tn)− δ

]+. (29)

Compare (28) and (29) to Lindley’s difference equations for the workload in a G/GI/1queue, with the beginning of an on-time corresponding to an arrival epoch. We im-mediately recognize that, by restricting our attention to off times alone, Y1(O(t)) is theworkload in an M/M/1 queue with arrival rate qu and mean service-time (λ1−g1)/g1qd,where O(t) is the total off time in [0, t]. From figure 1, it is also immediate that thebusy-period of source 1 is λ1/(λ1 − g1) times the busy-period of the M/M/1 queue.If we denote the LST of the busy-period of the M/M/1 queue, then the busy-periodLST of source 1 is given by

B(s) = BM

(λ1s

λ1 − g1

). (30)

Having characterized the busy-period of source 1, we now turn our attentionto buffer 2 whose asymptotic behavior we would like to characterize. Based on ouranalysis of the busy-period of source 1, source 2 can be thought of being served by aserver that alternates between an up and a down state as follows:

• Up state: Server stays in this state for an exponentially distributed duration 1/quand the server capacity is c in this state.

• Down state: Duration in this state has an LST B(s), and the server capacity is g2

in this state.


Now, define Y2(t) ≡ X2(t)/(c − λ2). Let F (y) be the steady-state cdf of of Y2,Fu(y) be the steady-state P (Y2 < y | server is up) and Fd(y) be the steady-stateP (Y2 < y | server is down). In a manner similar to the busy-period analysis forbuffer 1, it is easy to see that F c

u(y) = F cw(y), where Fw(y) is the waiting-time cdf in

an M/GI/1 queue with arrival rate qu, and service-time distribution LST B(s) given by

B(s) = B

(λ2 − g2

c− λ2s

), (31)

and the superscript c denotes ccdf. Thus, the LST of F cu(y), denoted by f c

u(s), canbe written as f c

u(s) = (1 − fu(s))/s, where fu(s) is given by the Kendall functionalequation for the busy-period of an M/GI/1 queue

fu(s) = B(s+ qu − fu(s)

). (32)

Now, let F cu(y) = P (Y2 > y and server is up) and F c

d (y) = P (Y2 > y and serveris down) in steady-state. Equating probability drifts, which can be justified rigorouslyusing level-crossing analysis, it is easy to see that

F cu(y) =

λ2 − g2

c− λ2F cd (y).

Therefore,

F c(y) = F cu(y) + F c

d (y) =c− g2

λ2 − g2F cw(y) =

(c− g2

λ2 − g2

)γF c

u(y), (33)

where

γ ≡ 1− ququ + qd

λ1

g1

is the probability that the server is up.Now it is easy to see that the F c(y) ∼ αy−3/2 e−δy as follows: The asymptotic

form of F c(y) is the same as that of F cu(y) except for a constant. Since F c

u(y) has thesame LST as that of the busy-period of an M/M/1 queue, from [3], its asymptotic formis as desired. The exact values of α and δ can be calculated from [3, equation (4.1)]using (31)–(33).

4.2. Numerical results

We generate high and low MAP sources using a superposition of on–off sourceseach with an average rate of 0.0125 as in [11]. As in [11], the mean on and off timesare 436.36 and 4363.63, with the arrival rate during an on period being 0.1375. Thus,a high-priority mean arrival rate of 0.05 would mean that there are four independenthigh-priority MMPP sources. The service times are exponential with mean 1. Ofcourse, the arrival processes and service times could be more general, but the modelconsidered here is sufficient to illustrate the asymptotic behavior. We use the results in


Table 1Parameters of the tail asymptote for low-priority waiting times for various

values of arrival rates for high-priority and low-priority sources.

ρh ρl δ β α× 103

0.025 0.025 0.2322 −0.7170 5.90.025 0.0375 0.2356 −0.3710 3.20.0625 0.025 0.04 −1 2.50.0625 0.0375 0.0411 −0.845 30.0625 0.05 0.0425 −0.5 1.4

Table 2Parameters of the low-priority tail asymptote with ρh = 0.0125.

ρl δ β α

0.0125 0.399 −1.18 0.1480.025 0.4088 −0.38 0.03880.0375 0.4089 2.3 1.3 · 10−6

0.05 0.4197 4.85 9 · 10−10

the previous sections to calculate the exact tail probabilities and use the moment-basedtechnique in [1] to compute the asymptote. To ensure the accuracy of the numericalexamples, we use a large number of moments (>40) and also stop the computationonly when there is less than a 10% change in the computed parameters. For instance,if δ is computed to be 0.4, then the number of moments used in the computation wouldbe large enough to ensure that the value of δ changed by less than 0.01 in successivesteps of the iteration given in [1].

The parameters of the asymptote, α, β and δ are shown in table 1 for variousvalues of ρh and ρl. As can be seen from the table, β can be non-negligible, thus,leading to non-exponential asymptotics, in general. It is also interesting to note thatthe so-called effective bandwidth approximation, e−δT , does not change very much asρl is changed with ρh = 0.0625. However, the true asymptote changes significantlyas revealed by the different values for α and β.

In the M/G, G/1 model studied in [4], it was observed that, as the low-priorityarrivals increased, the value of β exhibited the following behavior: β is equal to −3/2till a threshold value of the low-priority arrival rate is reached. At this threshold, β isequal to −1/2 and above this threshold, β is equal to zero. We numerically study if thisbehavior holds with MAP arrival processes. We consider an example with the sametype of on–off sources as before. We set the high-priority arrival rate λh = 0.0125, andincrease the low-priority arrival rate. The results are presented in table 2. From table 2,we see that the behavior of the asymptote is different with general MAP arrivals thanwith Poisson arrivals: either the asymptotic form of the tail distribution is different orthe behavior of β is different. As an example, if the asymptotic form that has beenhypothesized is correct, then β can be positive with MAP arrival processes which isdifferent from the behavior with Poisson arrivals.


While the form of the asymptote is yet to be proven, due to the fluid exampleand the results in [4], it is interesting to verify how well the expression (27) performsas an approximation to the exact tail probability. The analysis of M/G/1 queues withpriorities in [4] suggests that, when β 6= 0, the asymptote may not lead to a goodapproximation of the exact tail probabilities. This is illustrated in figures 3, 4. The

Figure 2. Non-exponential asymptote (dashed) and exact low-priority waiting-time tail probability (solid)for ρh = 0.0125, ρl = 0.0375.




figures also show another important feature: even if the asymptote is valid, it need notbe an upper bound for the exact tail probability. However, in general, it is not alwaysoptimistic either. For instance, in the case of Poisson arrivals for both high-priorityand low-priority sources, the non-exponential asymptote in [4, table 14.2] provides aconservative approximation to the exact tail probability.

Figure 2 shows the plots of the exact tail and the asymptote for the case withρl = 0.0375. The asymptote increases slightly for small values of T and then decreasessince β > 0. Figure 2 again illustrates that the asymptote may not be very accurate evenwhen T is large enough such that the tail probability is small (around 10−7). Further,the tail probability estimate is optimistic, which may not be suitable for applications.This points to a need for further work on proving the exact form of the asymptote, andexploring other approximations (other than the asymptote) which are both accurate aswell as faster than computing the exact tail probability.

Appendix

Proof of theorem 6Our proof is a generalization of a similar proof for vacation models in [26].

Consider the Markov renewal process obtained by sampling the system at departureepochs. Let p(n,m) be the stationary probability vector at departure epochs, i.e., thejth component of p(n,m) gives the stationary probability that there are n high-priority


and m low-priority customers, and the arrival process is in phase j at a departureepoch. From [35], the 2-dimensional transform of p(n,m), denoted by P (z,w), isgiven by (10), where

• A(z,w) is the transform of the number of arrivals of each class (z denotes the highpriority and w the low priority) in a service-time,

• Q(w) is the stationary distribution of the number of low-priority packets in thesystem obtained by sampling the system just after departures which leave no high-priority packets in the system, and

• GH (w) is the transform of the number of low-priority packet arrivals in a high-priority busy-period.

From the stationary distribution obtained by sampling the system just after departures,we need to derive the stationary distribution of the queueing process. Using PASTAwe would then have the distribution of packets including the number of each class,seen by a low priority arrival.

Let Y (z,w) =∑∞

n=0

∑∞m=0 y(n,m)znwm be the transform of the stationary dis-

tribution of the number of packets of each class in the queue.Let y(n,m, t) be a a vector whose ith component is the probability that there

are n packets of high priority, m packets of low priority, and the phase of the in-put process is i, at time t, starting at some initial state at time 0. The initial stateis irrelevant because we will be considering only steady-state quantities and the keyrenewal theorem helps us remove the dependence on the initial state. Thus, we sup-press the initial state in the analysis that follows. Let Mn,m(t) denote the renewalfunction (here a vector) associated with the Markov regenerative process (with theabove initial state), i.e., the ith component of the vector Mn,m(t) is the mean numberof visits to the state (n,m) and phase i in [0, t). The following equations hold fory(n,m, t):

(i) n > 0, m > 0,

y(n,m, t)

=

∫ t

0dM0,0(u)

∫ t−u

0dPa(1, 0, v)Pa(n− 1,m, t− u− v)

(1−H(t− u− v)

)+

∫ t

0dM0,0(u)

∫ t−u

0dPa(0, 1, v)Pa(n,m− 1, t− u− v)

(1−H(t− u− v)

)+

n∑k=1

m∑l=0

∫ t

0dMk,l(u)Pa(n− k,m− l, t− u)

(1−H(t− u)

)+

m∑l=1

∫ t

0dM0,l(u)Pa(n,m− l, t− u)

(1−H(t− u)

); (A.1)


(ii) n > 0, m = 0,

y(n, 0, t)

=

∫ t

0dM0,0(u)

∫ t−u

0dPa(1, 0, v)Pa(n− 1, 0, t− u− v)

(1−H(t− u− v)

)+

n∑k=1

∫ t

0dMk,0(u)Pa(n− k, 0, t− u)

(1−H(t− u)

); (A.2)

(iii) n = 0, m > 0,

y(0,m, t)

=

∫ t

0dM0,0(u)

∫ t−u

0dPa(0, 1, v)Pa(0,m− 1, t− u− v)

(1−H(t− u− v)

)+

m∑l=1

∫ t

0dM0,l(u)Pa(0,m− l, t− u)

(1−H(t− u)

); (A.3)

(iv) n = 0, m = 0,

y(0, 0, t) =

∫ t

0dM0,0(u)Pa(0, 0, t− u), (A.4)

where Pa(k, l, t) is the matrix of probabilities of k high-priority arrivals, and l low-priority arrivals in (0, t). We will explain equation (A.1); the other equations arespecial cases. We implicitly take into account information of the state of the arrivalprocess by considering all quantities to be vectors or matrices. Equation (A.1) isobtained by conditioning on the last epoch of the Markov renewal process as fol-lows:

(i) The last epoch of the Markov renewal process is the vector (0, 0) and one of thetwo following events occur so that the state at time t is (n,m):

(a) An arrival of high priority occurs first; this packet gets into service andremains in service. During the service of this packet n − 1 high-prioritypackets and m low-priority customers arrive.

(b) An arrival of low priority occurs first; this packet gets into service and remainsin service. During the service of this packet n high-priority packets and m−1low-priority customers arrive.

(ii) The last epoch of the Markov renewal process is (k, l) with 0 < k 6 n, 0 6 l6 m. One of the high-priority packets (the first actually) gets into service andstays in service. During the service of this packet n− k high-priority customersand m− l low-priority customers arrive.

(iii) The last epoch of the Markov renewal process is (0, l) with 0 < l 6 m. The firstlow-priority packet gets into service and stays in service. During the service ofthis packet n high-priority customers and m− l low-priority packets arrive.


As in [26], taking limits as t →∞, and using the key renewal theorem, we getthe following equations for y(n,m) = limt→∞ y(n,m, t):

(i) n > 0, m > 0,

y(n,m) =P (0, 0)/T(−(Dh

0 − λlI)−1)

Dh1

∫ ∞0

Pa(n− 1,m, t)(1−H(t)

)dt

+ P (0, 0)/T(−(Dh

0 − λlI)−1)

λlI

∫ ∞0

Pa(n,m− 1, t)(1−H(t)

)dt

+n∑k=1

m∑l=0

P (k, l)/T

∫ ∞0

Pa(n− k,m− l, t)(1−H(t)

)dt

+m∑l=1

P (0, l)/T

∫ ∞0

Pa(n,m− l, t)(1−H(t)

)dt; (A.5)

(ii) n > 0, m = 0,

y(n, 0) =P (0, 0)/T(−(Dh

0 − λlI)−1)

Dh1

∫ ∞0

Pa(n− 1, 0, t)(1−H(t)

)dt

+n∑k=1

P (k, 0)/T

∫ ∞0

Pa(n− k, 0, t)(1−H(t)

)dt; (A.6)

(iii) n = 0, m > 0,

y(0,m) =P (0, 0)/T(−(Dh

0 − λlI)−1)

λlI

∫ ∞0

Pa(0,m− 1, t)(1−H(t)

)dt

+m∑l=1

P (0, l)/T

∫ ∞0

Pa(0,m− l, t)(1−H(t)

)dt; (A.7)

(iv) n = 0, m = 0,

y(0, 0) = P (0, 0)/T(−(Dh

0 − λlI)−1)

. (A.8)

The 2-dimensional transform of y(n,m) yields the desired expression for Y (z,w).

Proof of theorem 7As in the proof of theorem 6 we can follow a detailed cookie-cutting argument

for the evolution of the distribution of customers in the queue. The resulting equationsare exactly like equations (A.1)–(A.4) in the proof of theorem 6 with the definitionsof Pa(k, l, t) and Mn,m(t) modified to take into account the joint phase of the high-and low-priority arrival processes. Defining Pa(z,w, t) to be the (double) z-transformof Pa(k, l, t), we have that Pa(z,w, t) = eD(z,w)t. The Key renewal theorem gives alimit similar to that obtained in equations (A.5)–(A.8) with the changes given in thestatement of theorem 7. We only have to take into account the changed Pa(z,w, t).For the renewal function, the dependence on the initial state disappears in the limit


as a consequence of the Blackwell renewal theorem. The case of lower traffic beingPoisson is obtained as a special case with D(z,w) = Dh

0 − λlIh +Dh1 z + λlIhw.

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